Subword complexity and power avoidance

TL;DR

This paper explores the relationship between subword complexity and power avoidance in infinite words, identifying extremal examples like Thue-Morse and 1-2-bonacci words under various constraints.

Contribution

It systematically characterizes extremal subword complexities for power-free infinite words, revealing new insights into their structure and uniqueness.

Findings

Thue-Morse has minimal subword complexity among overlap-free binary words.

Twisted Thue-Morse has maximal subword complexity among overlap-free binary words.

1-2-bonacci has minimal complexity among symmetric square-free ternary words.

Abstract

We begin a systematic study of the relations between subword complexity of infinite words and their power avoidance. Among other things, we show that -- the Thue-Morse word has the minimum possible subword complexity over all overlap-free binary words and all $(\frac 73)$-power-free binary words, but not over all $(\frac 73)^+$-power-free binary words; -- the twisted Thue-Morse word has the maximum possible subword complexity over all overlap-free binary words, but no word has the maximum subword complexity over all $(\frac 73)$-power-free binary words; -- if some word attains the minimum possible subword complexity over all square-free ternary words, then one such word is the ternary Thue word; -- the recently constructed 1-2-bonacci word has the minimum possible subword complexity over all \textit{symmetric} square-free ternary words.